# Moving of spin up and spin down electrons in a magnetic field [closed]

Suppose we have an electron in a conducting space, and now we apply a magnetic field. Now according to the right hand rule, the electron will have a circular motion in the plane perpendicular to the filed direction.

Now the question is, if we take into account the spin of electron, would the direction of the orbit be opposite for the two spins? If yes, why should it be so?

Considering only the spin, ignoring translational DOF, the Hamiltonian is $$H = -\mu ~\mathbf{B} \cdot \mathbf{S}$$

If $\mathbf{B}$ is directed along $z$, it's easy to see that $S_z$ states are energy eigentstates and thus are stationary.

Applying time evolution and taking the expectation value shows that if the spin is not oriented along $z$ initially, it will precess indefinitely in the plane normal to the field. However, if it is parallel (or antiparallel) to $\mathbf{B}$, it will stay that way. So if your spins are up ($+z$) and down ($-z$) initially, they will remain so and will not precess.

Now reintroducing the translational DOF, you've already identified that the motion in the plane is due to the Lorentz force, which only depends on the sign of the charge. Spin up / down with the same charge will have the same trajectories. So the answer to your question is no, the direction will be the same.

In a more interesting case, you could look at the time evolution of an electron with spin $\left|S_x; + \right>$ in a magnetic field oriented along $z$. I would expect that the electron's spin would rotate in $xy$ plane, while the electron also traces out a circular path in the plane (provided there's a central potential).

• I suggest that OP is confused in so far as to wonder whether the spin influences the spatial dynamics of the electron seen as a point particle. Commented Jun 24, 2016 at 13:41

Direction of deflection of electrons in magnetic field

It is not the full picture you are describing. A moving electron in a magnetic field gets deflected according the rule $\vec F = q \vec v \times \vec B$. This vector product has a direction and and this is what we observe in natur: all electrons get deflected in the same direction. Otherwise no electric drive nor electric generator would work.

Magnetic dipole moment and intrinsic spin

The deflection of electrons all in the same direction takes place due to the parallelity of the magnetic moment and the intrinsic spin. For macroscopic bodies you know this phenomenon as the gyroscopic effect. Positrons have a antiparallel alignement of spin and moment and get deflected in the oposite direction.

Pauli principle

The Pauli principle reflects our knowledge about electron states in atoms. It was found out that in every shaped region (s, px, py, pz, ...), called traditional electron orbitals, could be only two electrons and off opposite direction. But it is very important to notice that this two electrons are indistinguishable in free state, their intrinsic spin and their magnetic dipole moments are equally aligned.